A347395 Dirichlet convolution of Liouville's lambda (A008836) with A342001, where A342001(n) = A003415(n)/A003557(n).

0, 1, 1, 1, 1, 3, 1, 2, 1, 5, 1, 3, 1, 7, 6, 2, 1, 2, 1, 5, 8, 11, 1, 5, 1, 13, 2, 7, 1, 14, 1, 3, 12, 17, 10, 2, 1, 19, 14, 9, 1, 20, 1, 11, 5, 23, 1, 5, 1, 2, 18, 13, 1, 4, 14, 13, 20, 29, 1, 14, 1, 31, 7, 3, 16, 32, 1, 17, 24, 34, 1, 3, 1, 37, 3, 19, 16, 38, 1, 9, 2, 41, 1, 20, 20, 43, 30, 21, 1, 9, 18, 23, 32

Offset: 1

Author

Antti Karttunen, Sep 02 2021

Keywords

• nonn
• look

Comments

It seems that all the terms after the initial zero are strictly positive. Checked up to n = 2^24. Compare to A346485.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A003415, A003557, A008836, A342001, A347396 [= a(A276086(n))].

Cf. also A346485, A347235.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A008836(n) = ((-1)^bigomega(n));
A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));

Formula

a(n) = Sum_{d|n} A008836(n/d) * A342001(d).

Sum_{k=1..n} a(k) ~ c * A065464 * Pi^4 * n^2 / 180, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, Mar 04 2023